Ferromagnetism and Superconductivity in Two-band Hubbard

Models

Kazuhiro Sano and Yoshiaki Ono∗

(Department of Physics Engineering)

(Received July 18, 2007)

AbstractWe investigate ferromagnetism and superconductivity in two types of two-band Hubbard mod-

els: the multi-orbital Hubbard model and the d-p model, with particularly paying attention toeffect of the interplay between the electron correlation and the band splitting. As strong cor-relation plays a crucial role for these phenomena, we employ two complementary approaches:the numerical diagonalization method for one-dimensional models and the dynamical mean-fieldtheory for the infinite-dimensional model. These approaches give us reliable results beyond theperturbative approximation or the usual mean-field like approximation. For the one-dimensionalmodels, we calculate the critical exponent Kρ based on the Tomonaga-Luttinger liquid theoryand determine phase diagram of superconducting(SC) region by the condition Kρ > 1. The SCregion appears near partially polarized ferromagnetic region in the multi-orbital Hubbard modelwith finite band splitting. Analysis of paring correlation functions for the ground state indicatesthat the triplet paring is relevant to the superconductivity. In the one-dimensional d-p model, wefind enhancements of the singlet and triplet paring correlations at the SC region. Although onlythe singlet paring is relevant to superconductivity near half-filling, the triplet paring increaseswith electron filling and is relevant to the superconductivity as well as the singlet paring nearfull-filling. In the infinite-dimensional d-p model, we calculate the local green’s function on thebasis of the dynamical mean-field theory, where the effective impurity Anderson model is solvedusing the numerical diagonalization method. It shows the phase diagrams of the metal-insulatortransition on the ground state at half-filling and quarter-filling as functions of the Coulomb inter-action and the band splitting. We obtain the transition temperatures for the ferromagnetism andthe superconductivity as functions of the Coulomb interaction, the band splitting and the electronfilling. It indicates that the transition temperature of the triplet superconductivity is higher thanthat of the singlet superconductivity near the ferromagnetic state. These results suggest a closerelationship between the ferromagnetism and the superconductivity as a common feature of thetwo-band Hubbard models.

1 IntroductionThe strongly correlated electron systems with multi-band (multi-orbital) have attracted much

interest due to various interesting phenomena such as colossal magnetoresistance in manganitesLa1−xSrxMnO3[1], triplet-pairing superconductivity in the ruthenate Sr2RuO4[2], metal-insulatortransition in alkali-doped fullerides AxC60[3] and one-dimensional superconductivity of CuO2 doublechain in Pr2Ba4Cu7O15−δ[4]. Among them, spin-state transition of cobalt oxides in La1−xSrxCoO3[5,6, 7] give a good example displaying the effect of multi-orbital. Since the Hund’s rule coupling andthe crystal-field splitting between the t2g and eg orbitals are close to each other, the spin state ofthe cobalt ion depends on temperature, doping concentration and crystal structure. With increas-ing temperature, the low-spin state (t62ge

0g, S = 0) of the Co3+ (3d6) ion gradually changes into an

intermediate-spin state (t52ge1g, S = 1) and/or a high-spin (HS) state (t42ge

2g, S = 2)[6, 7, 8, 9]. With Sr

doping, La1−xSrxCoO3 shows a spin-glass state for x < 0.18 and a ferromagnetism for x > 0.18[10, 11].Layered cobalt oxides such as NaxCoO2 also show a a variety of interesting properties as multi-

band systems. For example, large thermoelectric power is discovered in Na0.5CoO2[12]. Weak fer-romagnetism has been observed in Na0.75CoO2[13]. The discovery of the superconductivity[14] inNaxCoO2 · yH2O with Tc ≈ 5K for x ≈ 0.35 and y ≈ 1.3 has stimulated considerable attention onthese materials. It has been suggest that large degeneracy of electronic states due to a competition

∗Department of Physics, Niigata University

1

Res. Rep. Fac. Eng. Mie. Univ., Vol.32,pp. 1-23(2007)Original Paper

－1－

between the Hund’s rule coupling and the crystal-field splitting plays a key role in the electronic statesof NaxCoO2 as well as of La1−xSrxCoO3[15].

With the new findings of these interesting materials, theoretical studies on the interplay betweenCoulomb interactions including the Hund’s rule coupling and band splitting by crystal-field are highlydesirable. The orbitally degenerate Hubbard model, which can be regarded as a kind of multi-bandsystem, has been extensively investigated to clarify the effect of orbital degrees of freedom in thepresence of intra-atomic Coulomb interaction. Many authors[16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26]have studied the ferromagnetism of this model and revealed that Hund’s rule coupling plays a crucialrole in ferromagnetism.

On the other hand, multi-band Hubbard model, which includes the effect of band splitting explic-itly, has also been extensively studied for a long time. In particular, there has been much theoreticalinterest in the d-p model as a model of the copper oxide high-temperature superconductor. The modelcontains hopping tpd between the Cu (d-orbital) site and the O(p-orbital) site and strong repulsiveinteractions at the d and p site (Ud and Up, respectively). In addition, it can contain the nearest-neighbor d-p interaction Upd and/or hopping tpp between the nearest-neighbor p sites. It is widelyaccepted as a basic model describing the electronic structure of the Cu-O network. Many theoreticalstudies have been performed on this model to explain the superconductivity and/or the antiferro-magnetism of the copper oxides. A few work has been discussed the possibility of so called flat bandferromagnetism in a type of d-p model[27].

In the present work, we address the multi-orbital Hubbard model in one-dimension and the d-pmodel in one- and infinite-dimension, while particularly paying attention to the effect of the interplaybetween the Coulomb interactions and the band splitting ∆. At the point of view of ferromagnetismand superconductivity in itinerant electron systems, we discuss these two models in detail. As thestrong correlation effect plays crucial roles in these interesting phenomena, a nonperturbative andreliable approach is required. We employ the numerical diagonalization method for the multi-orbitalHubbard model and the d-p model with finite system sizes in one-dimension, and the dynamicalmean field theory for the infinite-dimensional d-p model. These approaches are complementary foreach other in dimensionality and expected to give us reliable results beyond the perturbative or themean-field like approximations.

Numerical diagonalization approach has already been applied for the multi-orbital Hubbard modelat ∆ = 0 case[17, 18]. Although the available system size is fairly small, the results are in good agree-ment with the strong coupling analysis[18] and the results from the density-matrix renormalization-group method[23]. To examine the superconductivity, we calculate the critical exponent of the correla-tion functions Kρ based on the Luttinger liquid theory[28, 29, 30, 31]. The reliability of this approachhas been extensively tested for various one-dimensional models such as the Hubbard model[32], thet-J model[33] and the U -V model[34]. We can thus expect that this approach is reliable for themulti-orbital Hubbard model and the d-p model as well.

In the limit of infinite dimensions, the self-energy becomes purely site-diagonal and the dynamicalmean field theory (DMFT) becomes exact. The local Green’s function is given by the impurity Green’sfunction of an effective single impurity Anderson model. This effective one-site problem is calculatedby the numerical diagonalization of the finite size clusters. By using this method, Momoi et al. studiedthe multi-orbital Hubbard model at ∆ = 0[17]. They show the phase diagram of the ferromagnetism,but superconductivity was not discussed there. By calculating the magnetization and the pairingsusceptibility at finite temperature, we show the transition temperatures for the ferromagnetism andthe superconductivity as functions of the Coulomb interaction, the band splitting and the electronfilling.

2 Luttinger Liquid RelationAt first, we briefly discuss a general argument for 1D-electron systems based on the Luttinger

liquid theory. In the Luttinger liquid theory[28, 29, 30, 31], an effective Hamiltonian of 1D models inthe Tomonaga-Luttinger regime is generally given by

H =vσ

2π

∫ L

0

dx[Kσ(∂xθσ)2 + K−1

σ (∂xφσ)2]+

vρ

2π

∫ L

0

dx[Kρ(∂xθρ)2 + K−1

ρ (∂xφρ)2], (1)

where vσ, vρ, Kσ and Kρ are the velocities and coupling parameters of spin and charge parts, respec-tively. According to the Luttinger liquid theory, some relations have been established as universal

2

Kazuhiro Sano and Yoshiaki Ōno

－2－

relations in one-dimensional single band models[30, 31]. In the model which is isotropic in spin space,the coupling constant Kσ is renormalized to unity in the low energy limit and the critical exponentsof various types of correlation functions are determined by a single parameter Kρ.

For single band model, it is predicted that SC correlation is dominant for Kρ > 1 (the correlationfunction decays as ∼ r

−(1+ 1Kρ

) in the Tomonaga-Luttinger (TL) regime and as ∼ r− 1

Kρ in the Luther-Emery (LE) regime), whereas the CDW or SDW correlations are dominant for Kρ < 1 (the correlationfunctions decay as ∼ r−(1+Kρ) in the TL regime and as ∼ r−Kρ in the LE regime). Here, the LE regimeis characterized by a gap in the spin excitation spectrum, while in the TL regime, the excitation isgapless[28, 31]. In the case of non-interacting fermion systems, the exponent Kρ is always unity. Thus,the effective interaction between quasi-particles is attractive for Kρ > 1 whereas that is repulsive forKρ < 1.

The critical exponent Kρ is related to the charge susceptibility χc and the Drude weight D by

Kρ =12(πχcD)1/2, (2)

with D = πNu

∂2E0(φ)∂φ2 , where E0(φ) is the total energy of the ground state as a function of magnetic

flux Nuφ[31]. Here, the flux is imposed by introducing the following gauge transformation: cmσ† →eimφ/Nac†mσ for an arbitrary site m. When the charge gap vanishes in the thermodynamic limit, theuniform charge susceptibility χc is obtained from

χc =4/Nu

E0(Ne + 2, Nu) + E0(Ne − 2, Nu)− 2E0(Ne, Nu), (3)

where E0(Ne, Nu) is the ground state energy of a system with Nu unit sites(cells) and Ne electrons.The filling n is defined by n = Ne/Nu, where Nu is the total number of unit cells or sites. The valuesof D and χc are calculated from the ground state energy of the finite size system through eq.(2) and(3).

It is noted that the situation is complicated for two-band models. When both bands have beenoccupied by electrons simultaneously, Fermi points appear in both bands. In this case, the electronicstate of the low energy is completely changed and we can not use the above relations[35, 36, 37, 38].On the other hand, if the electron density is sufficiently small and/or the band splitting is sufficientlylarge, electrons occupy only lower band. In this case, Fermi level exits in only lower band and the lowenergy effective Hamiltonian is considered to be equivalent to that of the single band model. Therefore,we can adopt the above Luttinger liquid relations of the single band model for even two-band model.

We numerically diagonalize the model Hamiltonian and obtain the value of Kρ from the groundstate energy of finite size systems using the standard Lanczos algorithm. In addition to the numericaldiagonalization method, we use mean-field(MF) approximation to calculate the ground state energyand obtain the approximate value of Kρ of the infinite system. This approximation has been confirmedto be reliable at least in the weak coupling regime[39].

3 Multi-orbital Hubbard ModelPrevious works[17, 18, 19, 20, 21, 22, 23, 24] have studied the multi-orbital Hubbard model in

the case of the crystal-field splitting ∆ = 0. In one dimension, some rigorous results are shown in thestrong coupling limit U → ∞: the ground state is a fully polarized ferromagnetism for 0 < n < 2except for n = 1 when U ′ and J = J ′ are positive and finite[21]. The numerical result suggests thatthe ferromagnetism is stable also for n = 1 in the strong coupling region[23]. In infinite dimensions,the dynamical mean-field theory shows the existence of the ferromagnetism in the same parameterregion observed in one dimension[22]. They revealed that Hund’s rule coupling J plays a crucial rolein ferromagnetism. However, possible mechanisms of superconductivity and the effect of crystal-fieldsplitting ∆ was not considered in these studies.

In this section, we examine ferromagnetism and related superconductivity in the multi-orbitalHubbard model, while particularly paying attention to the effect of the interplay between J and∆[40, 41, 42].

3

Ferromagnetism and Superconductivity in Two-band Hubbard Models

－3－

3.1 Model HamiltonianWe consider the following Hamiltonian for the one-dimensional multi-orbital Hubbard model:

H = −t∑

i,m,σ

(c†i,m,σci+1,m,σ + h.c.) + U∑

i,m

ni,m,↑ni,m,↓

+ U ′∑

i,σ

ni,u,σni,l,−σ + (U ′ − J)∑

i,σ

ni,u,σni,l,σ +∆2

∑

i,σ

(ni,u,σ − ni,l,σ)

− J∑

i,m

(c†i,u,↑ci,u,↓c†i,l,↓ci,l,↑ + h.c.)− J ′

∑

i,m

(c†i,u,↑c†i,u,↓ci,l,↑ci,l,↓ + h.c.), (4)

where c†i,m,σ stands for the creation operator of an electron with spin σ in the orbital m (= u, l) atsite i and ni,m,σ = c†i,m,σci,m,σ. Here, t represents the hopping integral between the same orbitals andwe set t = 1 in this study.

The interaction parameters U , U ′, J and J ′ stand for the intra- and inter-orbital direct Coulombinteractions, the exchange (Hund’s rule) coupling and the pair-transfer, respectively. ∆ denotes theenergy difference between the two atomic orbitals, that is, crystal-field splitting. For simplicity, weimpose the relations, J = J ′ and U = U ′ + 2J ; the latter holds exactly in 3d-orbitals for ∆ = 0 andis a good approximation for ∆ 6= 0. The model in eq. (4) is schematically represented by Fig. 1(a).We numerically diagonalize the above model Hamiltonian up to 9 sites. In the noninteracting case(U = U ′ = J = 0), the Hamiltonian eq. 4 yields a dispersion relation ε±(k) = −2t cos(k)± ∆

2 , wherek is the wave vector and ε+(k) (ε−(k)) represents the upper (lower) band energy. This band structureis schematically represented by Fig. 1(b).

U’J

U

∆

t

t

J’Ek

F

ε+(k)

ε−(k)

∆

(a) (b)

Figure 1: Schematic diagrams of (a) the model Hamil-tonian and (b) the band structure in the noninteractingcase.

Figure 2: Schematic diagram ofMoebius boundary condition at∆ = 0.

3.2 Result for degenerate caseWe first exhibit the result at ∆ = 0 and compare it with the previous works. At ∆ = 0, two

bands are doubly degenerate and the finite size effect of the ordinary periodic boundary condition isfairly large. To reduce it, we use the Moebius boundary condition, which has been already used int-J ladder models[43]. The Moebius boundary condition as shown in Fig.2, is a kind of the periodicboundary condition and efficiently reduces the finite size effect of the systems than the ordinaryperiodic boundary condition.

Figure 4 shows the phase diagram of the ground state for the Nu = 8 system at n = 0.5 (4elec-trons/8sites) with the result of the DMRG method by Sakamoto et al.[23] on the U ′ vs. J parameterplane. A ferromagnetic ground state with full spin polarization(S = max) appears around J ' U ′

for U ′ >∼ 3. The parameter regions J/U ′ À 1 and J/U ′ ¿ 1 are paramagnetic(S = 0). No partiallypolarized state was found at this filling. Figure also shows that the difference between our result andthe DMRG result is small. The present result is similar to those by previous studies employing theperiodic or the anti-periodic boundary conditions, but the phase boundaries of their ferromagneticphases seem to depend on the system size greatly. We also indicate the phase boundary of the fer-romagnetic phase by using the mean-field(MF) approximation in Fig.4. In this approximation, theground-state energy E0 is calculated as

E0 = < H >

=∑

k<kF ,m,σ

ε±(k) + UNu

∑m

< nm,↑ >< nm,↓ > +U ′Nu

∑σ

< nu,σ >< nl,−σ >

+ (U ′ − J)Nu

∑σ

< nu,σ >< nl,σ > +∆2

Nu

∑σ

(< nu,σ > − < nl,σ >), (5)

4

Kazuhiro Sano and Yoshiaki Ōno

－4－

0 2 40.25

0.5

0.75

1

U’

Kρ

n=0.5∆=0

S=max

S=0 JcJc

J=U’J=U’/2

J=2U’

Figure 3: Kρ as a function of U ′ in thecases of J = U ′/2, J = U ′ and J =2U ′ at n = 0.5(4electrons/8sites). Jc

indicates the critical point of the singletground state changing into the fully polar-ized ferromagnetic(S=max) ground state.

0 5 10 150

5

10

15

U’

J

S=Max

n=0.5

S=0

∆=0

Sakamoto et al.MF

MF(ferro)

Kρ=0.5(Kρ<0.5)

SC

Figure 4: Phase diagram of the ferromag-netic state on the U ′−J parameter plane forn = 0.5(4electrons/8sites) at ∆ = 0. Thedashed line represents the phase boundaryof the ferromagnetic state obtained by MFapproximation. The solid triangle shows thepoint of Kρ = 0.5 on the U ′-axis.

where <> represents the expectation value of operators by the noninteracting ground state. Here, theterms −J

∑i,m(c†i,u,↑ci,u,↓c

†i,l,↓ci,l,↑ + h.c.) and −J ′

∑i,m(c†i,u,↑c

†i,u,↓ci,l,↑ci,l,↓ + h.c.) are omitted since

the expectation values of these terms are equal to zero. The result indicates that the ferromagneticphase appears all over the region except the near the origin. Comparing with the numerical results,it seems to overestimate the ferromagnetic region.

Figure 5 shows the value of Kρ as a function of U ′ for J = 0 at quarter filling n = 1(4electrons/4sites).We also show the result of the Green’s function Monte Carlo method obtained by Assaraf et al.[44]and the MF approximation for Kρ[39], which are represented by the broken and the dashed lines infig.5, respectively. Here, Kρ of the MF approximation is estimated by the Luttinger liquid relationas well as Numerical diagonalization method and the ground state is restricted to the singlet state,that is, the condition < nm,↑ >=< nm,↓ > is imposed. It shows good agreement with the numericalresult in the weak coupling regime, ensuring the small finite-size effect of the numerical calculation.As U ′ increases, Kρ decreases from unity to ∼ 0.42. For J = 0, the model Hamiltonian is equivalentto the SU(4) Hubbard model and the metal-insulator(MI) transition occurs at Kρ = 0.5[44]. WhenU ′ is larger than a critical value U ′

c ∼ 3, the MI transition is expected. Sakamoto et al. show then-dependence of the chemical potential at quarter filling and the existence of large charge gap inthe strong coupling region. Although the finite size effect does not allow the correct estimation ofthe transition point, they claim that the metal-insulator transition occurs at finite positive value ofU ′ = J ∼ 3.

Figure 6 gives the value of Kρ as a function of U ′ in several conditions of J = U ′/2, J = U ′ andJ = 2U ′ at n = 1. As U ′ increases, Kρ decreases, while it increases for a large U ′ in the conditionJ = 2U ′. In the region Kρ > 0.5, the SC correlation is expected to be the most dominant comparedwith the CDW and SDW correlations. To confirm the SC state, we calculate the lowest energy ofthe singlet state E0(φ) as a function of an external flux φ. As shown in Fig.7, the anomalous fluxquantization occurs in the region Kρ > 0.5, where the signature of it increases with J . It indicatesthat the SC state is surely realized in this parameter region.

In Figure 8, we give the phase diagram of the ground state for Nu = 4 at the electron densityn = 1(4electrons/4sites) with the result of DMRG method[23] on the U ′ vs. J parameter plane. Acomplete ferromagnetism with S=max appears around J ' U ′ in the strong coupling regime. Thisresult is in good agreement with the DMRG result obtained by Sakamoto et al.[23]. The dashed linepresents the boundary between the ferromagnetic state and the singlet state by using the mean-fieldapproximation. It seems to be fairly overestimate for the ferromagnetic region. It also indicates that

5

Ferromagnetism and Superconductivity in Two-band Hubbard Models

－5－

0 2 40

0.5

1

Kρ

U’

∆=0, J=0n=1

MF

Assaraf et al.

Figure 5: Kρ as a function ofU ′ in the cases of J = 0 forn = 1.0 (4electrons/4sites) at∆ = 0. The dashed line repre-sents a MF estimation for Kρ.The broken line indicates theGFMC result obtained by As-saraf et al.

0 2 4 60.25

0.5

0.75

1

U’

J=2U’

Jc

n=1∆=0

Kρ

J=0.5U’

J=U’S=maxS=0

Figure 6: Kρ as a function ofU ′ in the cases of J = U ′/2,J = U ′ and J = 2U ′ for n =1.0 (4electrons/4sites) at ∆ =0.

0

0.2

0.4

1.4

1.2

1.0

0.8

0.6

2π0

∆E

J=U’=0.0

∆=0

n=1

Figure 7: The energy differ-ence E0(φ) − E0(0) as a func-tion of an external flux φ atn = 1.0(6electrons/6sites) at∆ = 0.

the ferromagnetic region is smaller than the case of n = 0.5. In this ferromagnetic region, Sakamotoet al. also claimed that the system shows the triplet SC for J > U ′, while it becomes insulatorfor J < U ′. The SC phase boundary in the ferromagnetic region is smoothly connected to that inthe paramagnetic region as shown in Fig. 8. These results tell us that the Hund’s rule couplingJ plays important roles not only for the ferromagnetism but also for the superconductivity. In theparamagnetic state (S=0), we plot the phase boundary separating the SC region with Kρ > 0.5and the insulating region with Kρ < 0.5. As mentioned above, this SC phase is confirmed by theanomalous flux quantization, while this quantization disappears in the insulating region with large U ′

(not shown). Within the bosonaization method, the SC state has been identified as the triplet SCwith spin gap[25]. However, very recent work claims that the relevant symmetry of the paring is not

0 5 10 150

5

10

15

U’

J

S=max

S=0

n=1

S=max

Kρ=0.5

Kρ<0.5

(SC)

Kρ>0.5

(INSULATOR)

(ferro SC)S=0

(INSULATOR)

(spin gapped)

(spin gapped)

MF

MF(Ferro)

(SC)

Figure 8: Phase diagram of the ferromag-netic state on the U ′ − J parameter planefor n = 1(4electrons/4sites) at ∆ = 0. Thedashed line represents the phase boundaryof the ferromagnetic state obtained by MFapproximation. The solid triangle shows thepoint of Kρ = 0.5 on the U ′-axis.

0 2 4 6

0.5

1

n=8/6

Kρ

S=maxJ=U’

U’

∆=0

S=0

Jc

J=0.5U’

J=2U’

Figure 9: Kρ as a function of U ′ for n = 4/3(8electrons/6sites) at ∆ = 0.

6

Kazuhiro Sano and Yoshiaki Ōno

－6－

′p−′ wave like triplet but ′dxy−′ like singlet by using the bosonaization method and DMRG methodwithin the model at U ′ = 0[26].

To examine the case of over the quarter-filling(n=1), we calculate the value of Kρ as a function ofU ′ in several conditions of J = U ′/2, J = U ′ and J = 2U ′ at n = 4/3 (8electrons/6sites). As shownin Figure 9, it seems to be similar to the case of (n ≤ 1. This result suggests that the overall aspectof the electronic state dose not much depend on n.

3.3 Case for finite ∆Next, we consider the case of ∆ > 0. When the lowest energy of the upper band, ε+(0), is larger than

the Fermi energy, EkF , electrons occupy only the lower band with kF = πn2 and the model is regarded

as a ”single component” electron system. Hereafter, we mainly treat the case with ε+(0) > EkF . Inthis case, we can not adopt the Moebius boundary condition, since the symmetry of inter-band isbroken by finite ∆. In the case n < 1, we find the following boundary conditions are suitable, thatis, the periodic(antiperiodic) boundary condition for the lower(upper) band at Ne = 4m + 2 and theantiperiodic(periodic) boundary condition for the upper(lower) band at Ne = 4m, where Ne is thetotal electron number and m is an integer. This boundary condition can be regard as an extension ofthe Moebius boundary condition for the finite ∆.

For n > 1, on the other hand, our experience indicates that the usual boundary conditions, that is,the periodic boundary condition at Ne = 4m+2 and the antiperiodic boundary condition at Ne = 4mfor the both bands, are little better. This choice of the boundary conditions is rather empirical, butit seems to lead better results.

0 1 2 3

0.75

1

1.25

1.5

J(=U’)

Kρ

n=2/3

∆=1.0

1.2S=2

1.61.8

1.4

S=1

ferro(S=2)

S=1S=1

MF

Figure 10: Kρ as a function of J(= U ′)for n = 2/3(6electrons/9sites) at ∆ =1.0, 1.2, 1.4, 1.6, and 1.8. The down arrowsindicate the critical points of J(= U ′). Thedashed line represents a weak coupling esti-mation for Kρ.

0 1

0

0.2

0.4

0.6

E(Φ

)−E

(0)

Φ/2π

J(=U’)=0.4

1.0

1.3

0.8

n=2/3

∆=1.2

Figure 11: The energy difference E0(φ) −E0(0) as a function of an external flux φfor J(= U ′) =0.4,0.8,1.0 and 1.3 at n =2/3(6electrons/9sites) and ∆ = 1.2.

Figure 10 shows the value of Kρ as a function of J(= U ′) for several values of ∆ at the electrondensity n = 2/3(6electrons/9sites). As J increases, Kρ decreases for a small J , while it increases fora large J , and then becomes larger than unity. Since we consider the single component system, theSC correlation is expected to be the most dominant compared with the CDW and SDW correlationsin the region Kρ > 1. To confirm the superconducting state, we calculate the lowest energy of thesinglet state E0(φ) as a function of an external flux φ. As shown in Fig. 11, the anomalous fluxquantization occurs clearly at J ∼ 1.3, where Kρ is about 1.1. When J = 0.4, Kρ is less than unityand the anomalous flux quantization is not found.

When J is larger than a certain critical value, the ground state changes into the partially ferro-magnetic state with S=1 or S=2 from the singlet state. In Fig. 10, the broken line represents theMF approximation for Kρ. This approximation depends upon only the noninteracting ground statewhere the lower band is exclusively occupied, as shown in Fig. 1. Therefore, the effect of the upper

7

Ferromagnetism and Superconductivity in Two-band Hubbard Models

－7－

band is omitted and the result is independent of ∆. This approximation breaks down when the effectof the upper band becomes crucial. In this regime, Kρ rapidly increases with increasing J , and finallybecomes larger than unity indicating the superconducting state. The critical values of J with Kρ = 1are Jc ≈ 0.3, 1.0, 2.4 and 4.0 for ∆ = 1.0, 1.2, 1.6 and 1.8, respectively.

In Fig.12, we show the phase diagram of the superconducting state with Kρ > 1 together withthe partial ferromagnetic state with S = 1 and S = 2 on the U ′ vs. J parameter plane forn = 2/3(6electrons/9sites) at ∆ = 1.2, where the S = 2 state is depicted by the bunch of thinsolid circles(shadowed region). The superconducting phase appears near the partially polarized ferro-magnetic region. It extends from the attractive region with J < 0 and U ′ < 0 to the realistic parameterregion for 3d transition-metals with U ′ > J > 0. We have confirmed that the superconducting regionincreases as ∆ decreases, as shown in Fig.10.

Figure 13 shows the global phase diagram on the U ′ − J plane for n = 4/3(8electrons/6sites)at ∆ = 4.0. When U ′ >∼ J >∼ ∆, the fully polarized ferromagnetism with S = Smax appears. Itaccompanies the partially polarized ferromagnetism with 0 < S < Smax for J <∼ ∆. Inset shows themagnification of the phase diagram near the origin. As well as n < 1, the superconducting statewith Kρ > 1 appears near the partially polarized ferromagnetic region and it extends to the realisticparameter region with U ′ > J > 0. The configuration of the partially polarized ferromagnetic stateand the superconducting state is similar to the case n < 1. When J exceeds ε+(0) − EkF , the inter-band excitation develops and the occupation number of the upper band increases, which results ina large orbital fluctuation accompanied by the fluctuation between the low-spin and the high-spinstates. The mechanism of superconductivity may be related to this orbital fluctuation. We note thatsuperconductivity is also observed for the J ′ = 0 (J 6= 0) case in contrast to the previous study[45, 46],where the pair-transfer J ′ is crucial to superconductivity. The existence of the partially polarizedferromagnetism for ∆ > 0 has been reported in a different type of two-band Hubbard model[40].

In Fig. 14, we show the doping dependence of the critical values of J for the superconductivityand the ferromagnetism at ∆ = 4 with J = U ′, where we use n=4/3, 10/7, 3/2, 8/5 and 12/7 systems.We determine the critical values on condition that Kρ > 1 for the superconductivity and S > 0 forthe ferromagnetism. Although the finite size effect is considerably large, the phase boundary for thesuperconductivity can be approximately given by the phenomenological equation, Jc = ε+(0)− EkF

.Here, ε+(0) − EkF

corresponds to the lowest energy of the single-particle excitation from the lowerband to the upper band. This equation was found to be a good approximation for various n > 1[41].

0 2 40

2

4

J

U’

Kρ<1

Kρ>1

partial ferro

S=2S=1

S=0

S=1

∆=1.2n=2/3SC

Figure 12: Phase diagram of the supercon-ducting state with Kρ > 1 and the ferromag-netic states with S = 1 and S = 2(shadowedregion depicted by the bunch of thin solidcircles) on the U ′ − J parameter plane forn = 2/3(6electrons/9sites) at ∆ = 1.2.

0 5 10 150

5

10

15

1 20

1

2ferro

Kρ=1

U’

J

Kρ=0.8SC(Kρ>1)

MF

S=2 S=1

S=0

U’

J

n=4/3

S=4

S=3S=2

S=1

S=0

S=0 Kρ<1

(max)

S=2

SC MF

∆=4

(Kρ>1)

Figure 13: Global phase diagram on the U ′ − Jplane for n = 8/6 at ∆ = 4.0. The dashedline indicates the phase boundary of the com-plete(partial) ferromagnetic state obtained bythe MF approximation. Inset shows the mag-nification of the phase diagram near the origin.

8

Kazuhiro Sano and Yoshiaki Ōno

－8－

1 1.5 20

1

2

n

U’(=

J)

7units6units5units4units

ferro

Kρ>1Kρ<1

Figure 14: Phase diagram on the n − Jplane ∆ = 4.0. The broken line repre-sents the phenomenological equation Jc =ε+(0) − EkF (see text) as a function ofn. The solid(dashed) line indicates thephase boundary of the complete(partial) fer-romagnetic state obtained by the MF ap-proximation.

Suu

∆

Su−u,Tu−u Sul

Tul

∆

Sl−l,Tl−l Sll

Figure 15: Schematic diagrams of varioustypes of superconducting paring; Sll, Suu,Sl−l, Su−u and Sul( singlet paring) andTl−l(r), Tu−u(r) and Tul(r) (triplet paring)

We also show the phase boundary of the ferromagnetic state obtained by the MF approximation. Itseems to consist with the numerical result.

3.4 Paring correlation functionsNext, we consider superconducting paring of the system, as shown in Fig. 15. In Fig.16, we show var-

ious types of superconducting paring correlation functions C(r) in detail for n = 2/3(6electrons/9sites)at ∆ = 1.0 and J(= U ′) = 1.04. The paring correlation functions are defined by

Sll(r) =1

Nu

∑

i

< c†i,l,↑c†i,l,↓ci+r,l,↓ci+r,l,↑ >, (6)

Suu(r) =1

Nu

∑

i

< c†i,l,↑c†i,u,↓ci+r,u,↓ci+r,u,↑ >, (7)

Sl−l(r) =1

2Nu

∑

i

< (c†i,l,↑c†i+1,l,↓ − c†i,l,↓c

†i+1,l,↑)(ci+r+1↓ci+r,l,↑ − ci+r+1,l,↑ci+r,l,↓) >, (8)

Su−u(r) =1

2Nu

∑

i

< (c†i,u,↑c†i+1,u,↓ − c†i,u,↓c

†i+1,u,↑)(ci+r+1,u,↓ci+r,u,↑ − ci+r+1,u,↑ci+r,u,↓) >,(9)

Sul(r) =1

2Nu

∑

i

< (c†i,l,↑c†i+1,u,↓ − c†i,l,↓c

†i+1,u,↑)(ci+r+1,u,↓ci+r,l,↑ − ci+r+1,u,↑ci+r,l,↓) >, (10)

Tl−l(r) =1

2Nu

∑

i

< (c†i,l,↑c†i+1,l,↓ + c†i,l,↓c

†i+1,l,↑)(ci+r+1↓ci+r,l,↑ + ci+r+1,l,↑ci+r,l,↓) >, (11)

Tu−u(r) =1

2Nu

∑

i

< (c†i,u,↑c†i+1,u,↓ + c†i,u,↓c

†i+1,u,↑)(ci+r+1,u,↓ci+r,u,↑ + ci+r+1,u,↑ci+r,u,↓) >,(12)

Tul(r) =1

2Nu

∑

i

< (c†i,l,↑c†i+1,u,↓ + c†i,l,↓c

†i+1,u,↑)(ci+r+1,u,↓ci+r,l,↑ + ci+r+1,u,↑ci+r,l,↓) >, (13)

9

Ferromagnetism and Superconductivity in Two-band Hubbard Models

－9－

where C(r) = Sll(r), Suu(r), Sl−l(r), Su−u(r) and Sul(r) denote the singlet paring correlation functionson the same site in the lower orbital, on the same site in the upper orbital, between the nearest neighborsites in the lower orbital, between the nearest neighbor sites in the upper orbital, between lowerand upper orbitals on the same site, respectively. Further, Tl−l(r), Tu−u(r) and Tul(r) are the tripletparing correlation functions between the nearest neighbor sites in the lower orbital, between the nearestneighbor sites in the upper orbital and between lower and upper orbitals on the same site, respectively.The absolute value of Tu−u(r) is small, but the correlation of it is the slowest to decay. This result

1 2 3 4 5

10−6

10−4

10−2

n=6/9 ∆=1.0

|C(r

)|

r

Sll

Sl−l

Sul

Suu

Su−u

Tl−l

Tul

Tu−u

J(=U’)=1.04

Figure 16: The singlet paring correlationfunctions C(r) = Sll(r), Sl−l(r), Suu(r),Su−u(r), Sul(r) and the triplet correlationfunctions Tl−l(r), Tu−u(r), Tul(r), respec-tively(see text). Here we show the ab-solute value of the correlation functionsat ∆ = 1 and J(= U ′) = 1.04 forn=2/3(6electrons/9sites).

100

101

1 2 3 4 5 6100

101

102

103

Su−u

Sul

Sl−l

∆=1.0

|R(r

)|

Suu

Sll

|R(r

)| ∆=1.0

Tu−u

Tul

Tl−l

r

r0.42

Figure 17: The ratio of the sin-glet paring correlation functionsR(r) = C(r)J=1.04/C(r)J=0.2 for Sl−l(r),Suu(r), Su−u(r), Sul(r) and that of thetriplet correlation functions for Tl−l(r),Tu−u(r), Tul(r) with the power-low r0.42,respectively(see text). The broken linerepresents the power-low r0.42 predicted bythe Luttinger liquid relation.

seems to suggest that the relevant paring of the superconductivity is the tripletparing between lowerand upper orbitals on the same site and the ferromagnetic fluctuation near the ferromagnetic phasemay cause the paring. To clarify the behavior of the correlation functions, we calculate the ratio R(r)of the paring correlation functions at J(= U ′) = 1.04 and that of J(= U ′) = 0.2 as

R(r) =C(r)J=1.04

C(r)J=0.2. (14)

Although the correlation function C(r) decays as distance r increases, the function R(r) for relevantparing is expected to increase with r, because the value of Kρ at J = 1.04 is larger than that atJ = 0.2, where Kρ is about at 1.4 and 0.98, respectively. Then, the behavior of R(r) is expectedto ∼ r0.42. In fig.17, we show R(r) for Sll(r), Sl−l(r), Suu(r), Su−u(r) and Sul, (upper panel) andthe triplet paring correlation functions Tl−l(r), Tu−u(r) and Tul(r) with the power-low r0.42 predictedby the Luttinger liquid relation (lower panel), respectively. It indicates that that function R(r) forthe triplet paring, Tu−u, is much enhanced for long range paring correlation. On the other hand, theremains are not enhanced or decay as r increases. These results suggest that the paring correlationfunction Tu−u(r) is most relevant paring to the superconductivity. Although the system size is toosmall to compare the slope of the function R(r) with the power-low enhancement ∼ r0.42 directly, thebehavior of R(r) for Tu−u(r) seems to be roughly consistent with the result of the Luttinger liquidrelation.

4 d-p Chain ModelIn the previous works[47, 48, 49, 50, 51, 52], the present authors and many other authors studied

the one-dimensional d-p model with large on-site Coulomb repulsion Ud at Cu sites and intersite re-

10

Kazuhiro Sano and Yoshiaki Ōno

－10－

Ud

d∆

tpp

tpd

p

(a) (b)

ε+(k)

ε−(k)

∆

Figure 18: Schematic diagrams of (a) the model Hamiltonian and (b) the band structure in thenoninteracting case.

pulsion Upd, by using the numerical method. They claimed that the superconducting (SC) correlationis found to be dominant compared with the charge density wave (CDW) and spin density wave (SDW)correlations and the parameter Upd is central in inducing a SC state. However, we found that if thehopping term tpp is added to the model, the situation will be completely changed[53]. It enhances thefluctuation of charge and spin, and increases the exponent Kρ as parameter Upd does. Therefore, therepulsive interaction Upd is not always necessary for the SC state.

In this section, we reexamine the one-dimensional d-p model in the presence of the hopping termtpp to clarify the electronic structure of the SC state, especially paying attention to the symmetry ofthe paring correlation.

4.1 Model HamiltonianWe consider the following model Hamiltonian for the Cu-O chain in the hole picture;

H = tpd

∑

<ij>,σ

(p†iσdjσ + h.c.) + tpp

∑

<ij>,σ

(p†iσpjσ + h.c.)

+ εd

∑

j,σ

d†jσdjσ + εp

∑

i,σ

p†iσpiσ + Ud

∑

j

ndj↑ndj↓, (15)

where d†jσ and p†iσ stand for creation operators of a hole with spin σ in the Cu(d) orbital at site j andof a hole with spin σ in the O(p) orbital at site i, respectively, and ndjσ = d†jσdjσ. Here, tpd standsfor the transfer energy between the nearest-neighbor d and p sites and will be set at unity (tpd=1)hereafter in this study. The atomic energy levels of d orbital and p orbital are given by εd and εp,respectively. The charge-transfer energy ∆ is defined as ∆ = εp − εd.

For the noninteracting case (Ud = 0), the Hamiltonian in eq. (15) is easily diagonalized to yield adispersion relation

E±(k) =12

{εd + εp + 2tpp cos k ±

√(∆ + 2tpp cos k)2 + 16(tpd cos (k/2))2

}, (16)

where k is a wave vector and E+(k)(E−(k)) represent an upper (lower) band energy. For tpp > 0, thewidth of the lower band E−(k) decreases with decreasing ∆ and becomes perfectly flat at ∆flat =2tpp − t2pd/tpp. When ∆ < ∆flat, the band bends with a peak at k = 0. To investigate the electronicstructure of the interacting case, we numerically diagonalize the Hamiltonian up to 14 sites (7 unitcells). To carry out a systematic calculation, We use the periodic boundary condition for Nh = 4m+2and the antiperiodic boundary condition for Nh = 4m, where Nh is the total hole number and m isan integer.

4.2 Critical exponent Kρ

In Fig. 19, we show the numerical results of Kρ obtained through eq. 2 as a function of Ud forn = 4/3(8holes/6units) at ∆ = 2 and tpp = 0.5. We also plot Kρ obtained through the mean field(MF) approximation, respectively. In the MF approximation, the renormalized bands E±(k) are givenby eq. 16 in which ∆ is replaced by

∆ = ∆− 12Ud < nd >, (17)

where < nd > is the hole density at a d site and is determined by solving the self-consistent equation inthe MF approximation. Using the renormalized band E−(k), we can calculate the charge susceptibility

11

Ferromagnetism and Superconductivity in Two-band Hubbard Models

－11－

0 2 4 60

1

2

0 1

0

0.01

0.02

0.03

Ud=6.7

Eg(

Φ)−

Eg(

0)

Φ/2π

Ud=6.0

Ud

Kρ

∆=2, tpp=0.5n=8holes/6units

MF

Figure 19: The critical exponent Kρ as afunction of Ud at ∆ = 2 and tpp = 0.5 forn = 4/3(8holes/6units). The dashed linesrepresent the results of the MF approxima-tion (See in the text) Inset shows the energydifference E0(φ)−E0(0) as a function of anexternal flux φ

1.4 1.60

2

4

6

8

Ud

n

SC

Kρ=1

Kρ= 8

∆=2, tpp=0.5

Figure 20: The SC region sandwiched be-tween the two lines of Kρ = 1 and Kρ =∞ on the n-Ud parameter plane at ∆ =2 and tpp = 0.5. Here, we use n =8/6, 10/7, 6/4, 8/5, 10/6 and 12/7 systems.

χc and the Drude weight D. Substituting those values into eq. (2), we obtain Kρ within the MFapproximation. In the weak coupling regime, the results of the numerical diagonalization are in goodagreement with the MF approximation. It also seems to be roughly consistent with the numericalresults even in the strong coupling regime except for Ud

>∼ 6.When Ud

<∼ 3, Kρ decreases as Ud increases. For sufficiently large Ud, Kρ increases with increasingUd and diverges at Ud ∼ 6.8 when the charge susceptibility diverges. The region where Kρ is larger thanunity appears at 6.5 <∼ Ud

<∼ 6.8. When Kρ > 1, the SC correlation is expected to be most dominantcompared with the CDW and SDW correlations. Inset shows the energy difference E0(φ)− E0(0) asa function of an external flux φ. It shows that the anomalous flux quantization occurs at Ud = 6.7,when Kρ is about 2.0. While, at Ud = 6.0, Kρ ' 0.86 and the anomalous flux quantization is notfound. These result also confirms the SC phase at Ud = 6.7. The divergence of Kρ suggests thatthe effective bandwidth is close to zero. In fact, if we use the numerical value of < nd >' 0.83 withUd = 6.7 in eq. (7), we obtain ∆ ∼ −0.8 and the effective bandwidth is nearly equal to 0.1. It isvery small compared with the noninteracting band. The result of the anomalous flux quantizationshows that the variation of |E0(φ)−E0(0)| at Ud = 6.7 is much smaller than that at Ud = 6.0. It alsoindicates that the effective bandwidth is small.

Figure 20 shows the SC region where 1 < Kρ < ∞ on the n − Ud parameter plane at ∆ = 2 andtpp = 0.5. We use n = 8/6, 10/7, 6/4, 8/5, 10/6 and 12/7 systems, where numerator and denominatecorrespond to numbers of electrons and unit cells, respectively. The region sandwiched between thetwo lines of Kρ = 1 and Kρ = ∞ corresponds to the SC region. It is narrow near n = 1.4, however,it increases with increasing n.

4.3 Paring correlation functionsWe investigate various types of superconducting paring in detail. as shown in Fig. 21. Figure 22

shows the paring correlation functions C(r) defined by

Sp−on(r) =1

Nu

∑

i

< p†i↑p†i↓pi+r↑pi+r↓ >, (18)

Sd−on(r) =1

Nu

∑

i

< d†i↑d†i↓di+r↑di+r↓ >, (19)

12

Kazuhiro Sano and Yoshiaki Ōno

－12－

d∆

p

d∆

p

Sd−p, Td−p

Sp−p, Tp−p

Sd−d, Td−d Sd−on

Sp−on

Figure 21: Schematic diagrams of variousparing for d-p model.

10−4

10−2

1 2 3 4 5

10−4

10−2 Tp−p

Td−d

Td−p

∆=2, tpp=0.5Ud=6.7

Sp−on

Sd−on

Sp−p

Sd−d

Sd−p

∆=2, tpp=0.5Ud=6.7

r

|C(r

)||C

(r)|

Figure 22: The singlet paring correla-tion functions C(r) = Sp−on(r), Sp−p(r),Sd−on(r), Sd−d(r) and Sd−p(r) (upperpanel) and the triplet paring correlationfunctions Tp−p(r), Td−d(r) and Td−p(r)(lower panel), respectively (see text). Herewe show the absolute value of the correlationfunctions at ∆ = 2, tpp = 0.5 and Ud = 6.7for n=4/3(8holes/6units).

Sp−p(r) =1

2Nu

∑

i

< (p†i↑p†i+1↓ − p†i↓p

†i+1↑)(pi+r+1↓pi+r+1↑ − pi+r+1↑pi+r↓) >, (20)

Sd−d(r) =1

2Nu

∑

i

< (d†i↑d†i+1↓ − d†i↓d

†i+1↑)(di+r+1↓di+r+1↑ − di+r+1↑di+r↓) >, (21)

Sd−p(r) =1

2Nu

∑

i

< (d†i↑p†i↓ − d†i↓p

†i↑)(pi+r↓di+r↑ − pi+r↑di+r↓) >, (22)

Tp−p(r) =1

2Nu

∑

i

< (p†i↑p†i+1↓ + p†i↓p

†i+1↑)(pi+r+1↓pi+r+1↑ + pi+r+1↑pi+r↓) >, (23)

Td−d(r) =1

2Nu

∑

i

< (d†i↑d†i+1↓ + d†i↓d

†i+1↑)(di+r+1↓di+r+1↑ + di+r+1↑di+r↓) >, (24)

Td−p(r) =1

2Nu

∑

i

< (d†i↑p†i↓ + d†i↓p

†i↑)(pi+r↓di+r↑ + pi+r↑di+r↓) >, (25)

where Sp−on(r), Sd−on(r), Sp−p(r), Sd−d(r) and Sd−p(r) denote the singlet paring correlation func-tions on a same p site, on a same d site, between the nearest neighbor p sites, between the nearestneighbor d sites and between the nearest neighbor p and d sites, respectively. Further, Tp−p(r),Td−d(r) and Td−p(r) denote the triplet paring correlation functions between the nearest neighbor psites, between the nearest neighbor d sites and between the nearest neighbor p and d sites, respectively.Here, we show the absolute value of the correlation functions at ∆ = 2, tpp = 0.5 and Ud = 6.7 forn=4/3(8holes/6units). The orders of the absolute value of these correlation functions are almost equalto one another.

However, the correlation functions C(r) for Sp−on(r) and Sp−p(r) seem to decay slower thanthat of the remains. It suggests that these paring play an important role in the superconductivity.We also examine the ratio of the paring correlation functions R(r) at Ud = 6.7 and Ud = 6.0 asR(r) = C(r)Ud=6.7/ = C(r)Ud=6.0, where Kρ ∼ 2.0 at Ud = 6.7 and Kρ ∼ 0.86 at Ud = 6.0. Thefunction R(r) for relevant paring is expected to behave as r1.1. In fig.23, we show the ratio of theparing correlation functions R(r) for Sp−on(r), Sp−p(r), Sd−on(r), Sd−d(r) and Sd−p(r) with thepower-low r1.1 (upper panel) and that of the triplet paring correlation functions for Tp−p(r), Td−d(r)

13

Ferromagnetism and Superconductivity in Two-band Hubbard Models

－13－

0.6

1

2

1 2 3 4 50.6

2

Sp−p

Sd−pSd−d

∆=2, tpp=0.5

|R(r

)| Sp−on

Sd−onr1.1

|R(r

)|

∆=2, tpp=0.5

Tp−p

Td−pTd−d

Figure 23: The ratio of the sin-glet paring correlation functionsR(r) = C(r)Ud=6.7/C(r)Ud=6.0 forSp−on(r), Sp−p(r), Sd−on(r), Sd−d(r)and Sd−p(r) with the power-low r1.1

depicted the broken line (upper panel)and that of the triplet paring correlationfunctions for Tp−p(r), Td−d(r) and Td−p(r)(lower panel), respectively. Here we showthe absolute value of R(r) at ∆ = 2,tpp = 0.5 for n=4/3(8holes/6units).

10−2

10−1

100

1 2 3 4 5

10−1

100 Tp−p

Td−d

Td−p

1 2 3 4 51

2

|R(r

)|

r

Tp−p

Sp−pr0.15

Sp−on

Sp−p

Sd−d

Sd−on

Sd−p

|R(r

)||R

(r)|

Figure 24: The ratio of the sin-glet paring correlation functionsR(r) = C(r)Ud=6.59/C(r)Ud=4.14 forSp−on(r), Sp−p(r), Sd−on(r), Sd−d(r)and Sd−p(r)(upper panel), and that ofthe triplet paring correlation functionsfor Tp−p(r), Td−d(r) and Td−p(r) (lowerpanel), respectively. Here we show theabsolute value of R(r) at ∆ = 2, tpp = 0.5for n=12/7(12holes/7units). Inset showsR(r) for Sp−p(r) and Tp−p(r) with thepower-low r0.15.

and Td−p(r) (lower panel), respectively. It indicates that that R(r) for Sp−on(r) and Sp−p(r) aremuch enhanced at long range paring correlation and seem to be roughly consistent with the power-low obtained by the Luttinger liquid relation. On the other hand, the remains decay as distance rincreases or are almost constant. These results suggest that the singlet paring correlation functionsSp−on(r) and Sp−p(r) are most relevant to the superconductivity and the triplet paring is irrelevant.

On the other hand, the situation is changed for large hole density. In this case, the triplet par-ing correlation is enhanced as well as the singlet paring correlation. In fig.24, we show the ratioof the singlet paring correlation functions R(r) = C(r)Ud=6.59/C(r)Ud=4.14 for Sp−on(r), Sp−p(r),Sd−on(r), Sd−d(r) and Sd−p(r) (upper panel), and that of the triplet paring correlation functions forTp−p(r), Td−d(r) and Td−p(r)(lower panel), respectively, where we show the absolute value of R(r)for n=12/7(12holes/7units). Here, R(r) of the relevant paring is expected to behave as r0.15 sinceKρ ∼ 1.15 at Ud = 6.59 and Kρ ∼ 1.0 at Ud = 4.14. Inset shows R(r) for Sp−p(r) and Tp−p(r) withthe power-low r0.15. It shows that R(r) for Tp−p(r) increases with r as well as that for Sp−p(r), whichseems to be roughly consistent with the power-low r0.15 predicted by the Luttinger liquid relation.suggests that the triplet paring becomes relevant to the superconductivity as well as the singlet paringnear n = 2.

5 d-p Model in Infinite DimensionsRecently, some significant progress has been achieved in understanding the strongly correlated

electron systems by using the dynamical mean-field theory (DMFT)[54, 55]. In this approach, thelattice problem is mapped onto an effective impurity problem where a correlated impurity site isembedded in an effective uncorrelated medium that has to be determined self-consistently. To solvethe effective impurity problem, several methods have been applied including the iterated perturbationtheory[55], the non-crossing approximation[56], the quantum Monte Carlo (QMC) method[57], theexact diagonalization (ED) method[58] and the numerical renormalization group (NRG) method[59,60]. The DMFT becomes exact in the limit of infinite spatial dimensions (d = ∞)[54] and believed tobe a good approximation in high dimensions.

14

Kazuhiro Sano and Yoshiaki Ōno

－14－

In the d = ∞ single-band Hubbard model at half-filling, the Mott metal-insulator transition isfound to occur as a first-order phase transition at finite temperature below a critical temperatureTcr[55]. Below Tcr, a coexistence of the metallic and insulating solutions is found for the same valueof the on-site Coulomb interaction U in the range Uc1 < U < Uc2[55, 60]. The magnetic phasediagram was obtained as a function of doping, temperature and U . A commensurate antiferromagneticorder changes to an incommensurate state at a value of the doping which depends on U [61]. Theferromagnetism was observed for an intermediate interaction strength U >∼ 2 in non-bipartite latticessuch as the fcc-type lattice[62] but was completely suppressed for U <∼ 20 in bipartite lattices such asthe hypercubic lattice[63]. The superconducting phase is absent in the d = ∞ single-band Hubbardmodel[64].

The DMFT has also been applied to the multi-band Hubbard model to elucidate the effect of theHund’s rule coupling on the Ferromagnetism[20, 22] and the Mott transition[65, 66]. Several authorshave extensively studied the d-p model using the DMFT[58, 67, 68, 69, 70, 71, 72, 73, 74, 75]. TheMott transition was found to occur at n = 1 (or n = 3)[67, 68, 69, 70, 71, 73, 74], where n is thetotal electron number per unit cell and given by the sum of p- and d-electron numbers: n = np + nd.The phase diagram of the Mott transition was obtained over the whole parameter regime includingthe Mott-Hubbard type (U < ∆) and the charge-transfer type (U > ∆)[69, 70]. When the carrier isdoped with n > 1, the superconductivity was observed in the charge-transfer regime[58, 67, 69], whileit was not observed in the Mott-Hubbard regime. The antiferromagnetism was also observed at andnear n = 1[75]. More recently, the present authors discussed the ferromagnetism which was found tooccur at and near n = 2 in the intermediate interaction strength U ≈ 2∆ >∼ 2[72]. Therefore it isinteresting to discuss the relationship between the magnetism and the superconductivity in the d-pmodel in infinite dimensions and to obtain the phase diagram as functions of the parameters such asthe interaction, the electron filling and the temperature.

5.1 Model and formulationWe consider the d-p model on a Bethe lattice with infinite connectivity z →∞. The Hamiltonian

is written as

H =∑

i,j,σ

(ti,jd†iσpjσ + h.c.) + εp

∑

j,σ

p†jσpjσ + εd

∑

i,σ

d†iσdiσ + Ud

∑

i

ndi↑n

di↓, (26)

where d†iσ and p†jσ stand for creation operators of a electron (or a hole) with spin σ in the d-orbitalat site i and in the p-orbital at site j, respectively. nd

i,σ = d†iσdiσ. ti,j = tpd√z

represents the transferenergy between the nearest neighbor site and the parameter tpd will be set to unity in the presentstudy. The atomic energy levels of d-orbital and p-orbital are given by εd and εp, respectively. Thecharge-transfer energy ∆ is defined as ∆ = εp − εd > 0.

In the DMFT, the effective action of the impurity problem is given by

S = Ud

∫ β

0

dτnd↑(τ)nd↓(τ)−∫ β

0

∫ β

0

dτdτ ′∑

σ

d†σ(τ)D−10σ (τ − τ ′)dσ(τ ′), (27)

where the Weiss function D0σ includes effects of the interaction at all the sites except the impuritysite. This action is derived by tracing out the fermionic degrees of freedom in the original lattice modelexcept the impurity site. The local Green’s function Dσ(τ−τ ′) = −〈Tdσ(τ)d†σ(τ ′)〉S is calculated withthis action. Using D0σ and Dσ, we introduce the local self-energy Σσ(iωn) = D0σ(iωn)−1−Dσ(iωn)−1,where ωn is the Matsubara frequency, ωn = (2n + 1)π/β. The self-consistency condition for the localGreen’s functions gives the relations

Dσ(iωn) =∫

dεN(ε)× iωn + µ− εp

(iωn + µ− εd − Σσ(iωn))(iωn + µ− εp)− ε2, (28)

Pσ(iωn) =∫

dεN(ε)× iωn + µ− εd − Σσ(iωn)(iωn + µ− εd − Σσ(iωn))(iωn + µ− εp)− ε2

, (29)

where µ is the chemical potential and Pσ(τ − τ ′) = −〈Tpσ(τ)p†σ(τ ′)〉S is the local Green’s functionat p-site. For the Bethe lattice with z = ∞, the density of states is given by a semi-circular function,

15

Ferromagnetism and Superconductivity in Two-band Hubbard Models

－15－

N(ε) =√

4− (ε/tpd)2/2πtpd. Using the semi-circular density of states in eqs.(28) and (29) andeliminating Σσ(iωn), we obtain the simple form of the self-consistency equations

D0σ(iω)−1 = iωn + µ− εd − t2pdPσ(iωn), (30)

Pσ(iω)−1 = iωn + µ− εp − t2pdDσ(iωn). (31)

To calculate the local Green’s function Dσ(iωn) for a given D0σ(iωn), we use the numerical diag-onalization method (DMFT-ED method). The self-consistency equations (30) and (31) lead a newD0σ(iω) and we repeat the calculation of the local Green’s function. This process is iterated until thesolutions converge.

In the DMFT-ED method, we approximately solve the impurity Anderson model of a finite-sizecluster;

HAnd = ε0σ

∑σ

ndσ +Ns∑

l=2,σ

εlσc†lσclσ +Ns∑

l=2,σ

Vlσ(d†σclσ + c†lσdσ) + Udnd↑nd↓, (32)

where ε0σ is the impurity level and εlσ (l = 2, 3, ..., Ns) are levels of the ’conduction electron’ hybridizedwith the impurity by Vlσ. We regard the non-interacting Green’s function GAnd

0σ (iωn) as the Weissfunction D0σ(iω) in the action eq.(27). Then, the interacting Green’s function GAnd

σ (iωn) correspondsto the local Green’s function Dσ(iω) in the original lattice problem. Here, GAnd

0σ (iωn) is defined by

GAnd0σ (iωn) =

1

iωn − ε0σ −∑Ns

l=2

V 2lσ

iωn−εlσ

. (33)

For a given D0σ(iω), we determine 2Ns−1 parameters ε0σ, εlσ, Vlσ(l = 2, 3, ..., Ns) to make GAnd0σ (iωn)

as close to D0σ(iω) as possible. Using these parameters, we diagonalize the finite cluster of the impurityAnderson model, and calculate GAnd

0σ (iωn) (D0σ(iωn)). At finite temperature, the Green’s function isstraightforwardly calculated from the full set of states |i〉 with eigenvalues Ei according to

Dσ(iωn) =1Z

∑

i,j

|〈i|d†σ|j〉|2iωn − Ei + Ej

(e−βEi + e−βEj ). (34)

Using the Lanczos method with the continued-fraction expansions, we also calculate the zero tem-perature Green’s functions. In this case, we replace the Matsubara frequencies by a fine grid ofimaginary frequencies, β (iωn = (2n + 1)π/β), where the fictitious inverse temperature β determinesa low-frequency cut-off.

5.2 Metal-Insulator transition at n = 2First, we consider the paramagnetic state at zero temperature for n = nd + np = 2. In the

non-interacting case Ud = 0, the system is a band-insulator with the energy gap ∆ for ∆ 6= 0 whileit is a semimetal for ∆ = 0. Within the restricted Hartree-Fock (HF) approximation, the energy gapis given by ∆ − Udnd

2 . Then the system is metallic for Ud = 2∆, otherwise it is insulating. We notethat nd = np = 1 for Ud = 2∆ due to the particle-hole symmetry. In the inset in Fig. 25, we showthe chemical potential µ as functions of n for Ud = 7, 8, 9, 10 at ∆ = 6 and T = 0 calculated from theDMFT-ED method with the system size Ns = 8[72]. In this calculation, the solution is restricted tothe paramagnetic state with ε0σ = ε0, εlσ = εl and Vlσ = Vl for l = 2, 3, ..., Ns. When Ud increasesfrom Ud = 0, the discontinuity in the chemical potential at n = 2 decreases and finally becomes zeroat a critical value, where a transition from the band-insulator to the correlated semimetal occurs[67].The critical values for the metal-insulator transition are plotted in Fig. 25. In contrast to the HFapproximation, the metallic state is found in the wide parameter region due to a correlation effectconsidered in the DMFT.

5.3 FerromagnetismAt low temperature, the correlated semimetal mentioned above becomes unstable compared to a

ferromagnetic state. In Fig. 26, we plot the magnetization for the d-electron Md, that for the p-electronMp and the total magnetization M = Md +Mp as functions of the temperature T at n = 2 for Ud = 8

16

Kazuhiro Sano and Yoshiaki Ōno

－16－

0 5 100

5

10

1.95 2 2.05

−0.2

−0.1

0

µ

n

T=0, ∆=6

Ud=10

9

8

7

Ud

∆

Metal

InsulatorInsu

lato

r Ud=2∆T=0n=2

Figure 25: The phase boundary separating the metallic and insulating regimes as a function of Ud and∆ at n = 2 and T = 0. The inset shows the chemical potential as functions of n for Ud = 7, 8, 9, 10 at∆ = 6 and T = 0 calculated from the DMFT-ED method with Ns = 8.

and ∆ = 4 calculated from the DMFT-ED method with Ns = 6[72]. As shown in Fig. 26, Md and Mp

have opposite sign to each other. In the low temperature limit, both of Md and Mp become constant,while the sum of them M becomes zero. The feature of the ferromagnetism from the DMFT issimilar to that from the HF approximation as shown in Fig. 26. However, the transition temperatureTc from the HF approximation is much higher than that from the DMFT (see also Fig. 29). InFig. 27, we plot the magnetization as functions of Ud with keeping ∆ = Ud

2 at n = 2 and T = 0.01calculated from the DMFT-ED method together with those from the HF approximation. When Ud

increases, the each component of the magnetization monotonically increases. In both approximations,the ferromagnetism is observed for the intermediate interaction strength Ud

>∼ 2. This is a strikingcontrast to the single-band Hubbard model where the ferromagnetism is observed only for the strong

0 0.02 0.04

−0.2

0

0.2

0 0.1 0.2

M

T (DMFT)

n=2

DMFT(Ud=8)HF(Ud=2)

M=Md+Mp

Md

Mp

T (HF)

Figure 26: The magnetization for the d-electron Md, that for the p-electron Mp andthe total magnetization M = Md + Mp asfunctions of the temperature T at n = 2,obtained from the DMFT-ED method forUd = 8 and ∆ = 4 with Ns = 6 and from theHF approximation (dashed lines) for Ud = 2and ∆ = 1.

0 2 4 6 8−0.2

−0.1

0

0.1

0.2

−1

−0.5

0

0.5

1

M(D

MFT

)

Ud=2∆

M(H

F)

n=2

DMFTHF

Md

Mp

M=Md+Mp

T=0.01

Figure 27: The magnetization for the d-electron Md, that for the p-electron Mp andthe total magnetization M = Md + Mp asfunctions of Ud with ∆ = Ud/2 at n = 2and T = 0.01, obtained from the DMFT-ED method and from the HF approximation(dashed lines).

17

Ferromagnetism and Superconductivity in Two-band Hubbard Models

－17－

coupling region U >∼ 20 in bipartite lattices within the DMFT[63]. We note that the value of themagnetization from the DMFT is about five times smaller than that from the HF for the same valuesof Ud and ∆.

Fig. 28 shows the total magnetization M as a function of the electron filling n. When n decreases,M continuously becomes zero at a critical value of n for high temperatures (see for T = 0.025), while itdiscontinuously becomes zero for low temperatures (see for T = 0.01). The similar properties are alsoobserved within the HF approximation as shown in Fig. 28. At low temperatures, however, the HFapproximation also predicts a metastable state where M decreases with increasing n and continuouslybecomes zero at a critical n (see for T = 0.025). By calculating the thermodynamic potential, we findthat the phase separation of the ferromagnetic state and the paramagnetic state occurs at the lowtemperatures (see Fig. 29).

Fig. 29 shows the transition temperature for the ferromagnetism Tc as a function of n for severalvalues of Ud(= 2∆)[72]. Tc monotonically decreases with decreasing n. The closed circles showthe second-order phase transition, while the open circles show the discontinuous transition as seenin Fig. 28. Within the HF approximation, the second-order phase transition occurs at the hightemperature (solid line), while the phase separation occurs at the low temperature (area between thedotted lines). In the DMFT, it is difficult to calculate the thermodynamic potential directly fromthe local Green’s function. But we may expect that, within the DMFT, the similar phase separationtakes place at low temperatures where the magnetization shows a discontinuous transition as shownin Fig. 28.

5.4 SuperconductivityFinally, we discuss the superconductivity in the d-p model (26). In infinite dimensions, the on-site

paring susceptibility χ of this model is given by[55, 67]

χ =1N

∫ β

0

dτ∑

ij

< Tdi↑(τ)di↓(τ)d†j↓(0)d†j↑(0) >

= T∑

ν,ν′[α−1/2{I − Λ}−1 · Λ · α−1/2]ν,ν′ , (35)

1.6 1.8 20

0.1

0.2

0.3

0.4

M

n

T=0.025

T=0.01

DMFT(Ud=8)HF(Ud=2)

T=0.1

T=0.025

Figure 28: The total magnetization M as afunction of the electron number n, obtainedfrom the DMFT-ED method for Ud = 8 and∆ = 4 at T = 0.01 (closed circles), 0.025(open circles), and from the HF approxima-tion for Ud = 2 and ∆ = 1 at T = 0.025(dotted line), 0.1 (dashed line).

1.6 1.8 20

0.01

0.02

0.03

0.04

T c

n

Ud=8

Ud=4

Ud=2

Tc HF /5 (Ud=2)

PS

Figure 29: The transition temperature Tc

for the ferromagnetism as a function of theelectron filling n, obtained from the DMFT-ED method (closed and open circles) forUd = 2∆ = 8, 4, 2, and from the HF approx-imation (solid and dotted lines) for Ud =2∆ = 2.

18

Kazuhiro Sano and Yoshiaki Ōno

－18－

with

Λν,ν′ = t4pd|P (iν)|[χloc]ν,ν′ |P (iν′)|, (36)

αν,ν′ = t4pd|P (iν)|2δν,ν′ , (37)

where, χloc is the local paring susceptibility at a d-site given by

[χloc]ν,ν′ = T 2

∫ β

0

dτ1

∫ β

0

dτ2

∫ β

0

dτ3

∫ β

0

dτ4eiν(τ1−τ2)eiν′(τ3−τ4) < Td↑(τ1)d↓(τ2)d

†↓(τ3)d

†↑(τ4) > . (38)

To calculate χloc within the DMFT-ED method, we use a spectral representation of r.h.s. in eq.(38)by inserting a complete set of eigenstates |i〉.

When the largest eigen value of Λ, λmax, approaches unity, the paring susceptibility diverges. Itsignals the transition into the superconducting state from the normal state. In Fig.30, we show thevalue of λmax as a function of the temperature T for Ud = 8 and ∆ = 4 at n = 1.3 and 1.7 obtainedfrom the DMFT-ED with the system size Ns = 5. The value of λmax increases with decreasing Tand exceeds unity at a certain critical temperature. For n = 1.3, the critical temperature for thespin-singlet pairing, TSS , is higher than that for the spin-triplet pairing, TTS , while, for n = 1.7, TTS

is higher than TSS [76]. This tendency is consistent with the zero-temperature DMFT-ED result withthe system size Ns = 6− 8[58].

We note that the triplet superconductivity is on-site pairing and the gap function is odd in the Mat-subara frequency as first proposed by Berezinskii in the superfluid 3He[77]. In the present calculation,the eigen function corresponding to the largest eigen value of Λ for the triplet-pairing, is proportionalto the gap function at the critical temperature, and is confirmed to be odd in the Matsubara frequency.

In Fig.31, the superconducting transition temperature for the triplet pairing TTS and that for thesinglet pairing TSS , obtained from the DMFT-ED method mentioned above, are plotted as functionsof n for Ud = 8 and ∆ = 4. We also plotted the transition temperature for the ferromagnetismTc (see Fig. 29) together with that for the antiferromagnetism TN calculated from the DMFT-EDmethod with Ns = 6. As seen in Fig.31, the singlet superconductivity is realized for n <∼ 1.4 nearthe antiferromagnetic phase, where the antiferromagnetic fluctuation is considered to be responsiblefor the singlet pairing. On the other hand, the triplet superconductivity is realized for T >∼ 1.4 nearthe ferromagnetic phase, where the ferromagnetic fluctuation is considered to be responsible for thetriplet pairing. Significantly, a reentrant superconducting transition is observed for 1.4 <∼ n <∼ 1.5. Thereentrant transition has also been observed in the d = ∞ periodic Anderson model[78]. Therefore, the

0 0.02 0.040.5

1

1.5

λ max

T

Ud=8, ∆=4

singlet

triplet

n=1.7

n=1.3

Figure 30: The largest eigen value λmax asa function of the temperature T for Ud = 8and ∆ = 4 at n = 1.3 and 1.7 obtained fromthe DMFT-ED with Ns = 5.

1 1.5 20

0.01

0.02

0.03

0.04

T

n

Ud=8, ∆=4

TSSTN

Tc

TTS

Figure 31: Transition temperatures for theferromagnetism Tc, the spin-triplet super-conductivity TTS , the spin-singlet supercon-ductivity TSS and the antiferromagnetismTN , obtained from the DMFT-ED for Ud =8 and ∆ = 4.

19

Ferromagnetism and Superconductivity in Two-band Hubbard Models

－19－

reentrant superconducting transition is considered to be a specific feature of two-band type modelsincluding d-p model and the periodic Anderson model.

6 Summary and DiscussionWe have investigate the ferromagnetism and the related superconductivity of the Hubbard model

with two-fold orbital degeneracy and the d-p model with paying attention to the effect of the interplaybetween the Coulomb interactions and the band splitting. To obtain reliable results beyond theperturbative or the mean-field like approximations, we use the numerical diagonalization method forthe one-dimensional models and the dynamical mean-field theory for the infinite-dimensional model.For one-dimensional models, we calculate the critical exponent Kρ based on the Luttinger liquidtheory and the paring correlation functions of the ground state.

In the one-dimensional multi-orbital Hubbard model, we have obtained various phase diagramsincluding the ferromagnetic and the superconducting state on the U ′ − J parameter plane. Thefully polarized ferromagnetism has been found in the strong coupling regime with U ′ >∼ J >∼ ∆. For1 < n < 2, the ferromagnetism is metallic and mainly caused by the double-exchange mechanism[23].Crystal-field splitting destroys the fully polarized ferromagnetism resulting in a partially polarizedone for J <∼ ∆. In the vicinity of the partially polarized ferromagnetism, we have found the tripletsuperconducting phase, when J exceeds the lowest energy of the inter-band excitation. It extends tothe realistic parameter region for 3d transition-metals with U ′ > J .

Sakamoto et al.[23] claimed that the metallic ferromagnetism for ∆ = 0 appears in a similar pa-rameter region in any dimension by comparing the results from one dimension with those from infinitedimensions. It is natural to think that this ferromagnetism will appear in two and three dimensionseven in the presence of ∆. Then, we expect that a partially polarized (weak) ferromagnetism ap-pears in real materials, in which Hund’s rule coupling and crystal-field splitting compete with eachother, such as in cobalt oxides. In fact, a weak ferromagnetism has been observed in the layeredNa0.75CoO2[13] as well as in the perovskite R1−xAxCoO3[79]. The orbital degeneracy of 3d electronsis considered to play a crucial role in NaxCoO2 as well as in La1−xSrxCoO3[15].

In our calculation, we can not find any sign of the superconductivity near the fully polarizedferromagnetic state at the realistic parameter region U ′ > J . It suggests that not the completeferromagnetic state but the weak ferromagnetic state can be a key to the superconductivity of realmaterials. The competition between Hund’s rule coupling and crystal-field splitting causes the largeorbital fluctuation, accompanied by the fluctuation between the low-spin and the high-spin states ateach Co ion. This fluctuation is expected to mediate the superconductivity near the weak ferromag-netism. Since the orbital fluctuation has a local character, the mechanism for the superconductivitycould be common in all dimensions. We therefore think that exploration of the superconductivity inthe vicinity of the weak ferromagnetism in the perovskite R1−xAxCoO3[79] may be promising.

In the one-dimensional d-p model, we have calculated the critical exponent Kρ and the varioustypes of the paring correlation functions. Using the Luttinger liquid relations, we have found that theSC correlation is dominant in the parameter region where the renormalized band becomes almost flat.Since the narrowing of the effective bandwidth leads to the enhancement of the fluctuation, it can causethe superconductivity. The behavior of the paring correlation functions suggests that the singlet paringon a same p site and between the nearest neighbor p sites are most relevant to the superconductingstate near half-filling. This result can be interpreted that the antiferromagnetic fluctuation meditatesthe singlet paring as similar as the t-J model. However, in large doping case n = 12/7, the tripletparing between the nearest neighbor p sites becomes relevant to the superconductivity as well as thesinglet paring between the nearest neighbor p sites. This result suggests that the low energy physicsof the d-p model can not well described by the t-J model near n = 2. We think that the existence ofmulti-band might be crucial to understand the triplet paring superconductivity.

We also examine infinite-dimensional d-p model based on the dynamical mean-field theory. Thedynamical mean-field theory becomes exact in the limit of infinite spatial dimensions and believed tobe a good approximation in high dimensions. The result is expected to be complementary to the resultof the one-dimensional d-p model. We obtain the phase diagrams of the metal-insulator transition onthe ground state at half-filling and quarter-filling. We also calculate the magnetization and the pairingsusceptibility to obtain the transition temperatures for the ferromagnetism and the superconductivity.It shows that the singlet superconductivity is realized for n <∼ 1.4 near the antiferromagnetic phase,where the antiferromagnetic fluctuation is considered to be responsible for the singlet pairing. On

20

Kazuhiro Sano and Yoshiaki Ōno

－20－

the other hand, the triplet superconductivity is realized for T >∼ 1.4 near the ferromagnetic phase.We think that the ferromagnetic fluctuation is responsible for the triplet pairing as well as the one-dimensional case near n = 2.

Ferromagnetism and superconductivity has been studied for a long time as a central argument initinerant electron systems. In this paper, we have presented the electron correlation and the bandsplitting to be crucial for the ferromagnetism and the related superconductivity in the two types of thetwo-band Hubbard models. We hope that our work would yield an insight into the deep relationshipbetween these interesting phenomena.

References[1] For a review, see, for example, Y. Tokura, Ed., Colossal Magnetoresistive Oxides (Gordon and

Breach Science, New York, 2000).

[2] Y. Maeno, H. Hashimoto, K. Yoshida, S Nishizaki, T. Fujita, J. G. Bednorz and F. Lichtenberg,Nature (London) 372 532 (1994).

[3] O. Gunnarsson, E. Koch and R. M. Martin, Phys. Rev. B54 R11026 (1996).

[4] M. Matsukawa, Yuh Yamada, M. Chiba, H. Ogasawara, T. Shibata, A. Matsushita and Y. Takano,Physica C 411 101 (2004).

[5] G. H. Jonker and J. H. Van Santen, Physica 19 120 (1953).

[6] R. R. Heikes, R. C. Miller and R. Mazelsky, Physica 30 1600 (1964).

[7] P. M. Raccah and J. B. Goodenough, Phys. Rev. 155 932 (1967).

[8] K. Asai, P. Gehring, H. Chou, and G. Shirane, Phys. Rev. B40 10982 (1989).

[9] M. Itoh, M. Sugahara, I. Natori and K. Motoya, J. Phys. Soc. Jan. 64 3967 (1995).

[10] M. Itoh, I. Natori, S. Kubota and K. Motoya, J. Phys. Soc. Jpn. 63 1486 (1994).

[11] D. Louca, J. L. Sarrao, J. D. Thompson, H. Roder and G. H. Kwei, Phys. Rev. B60 10378 (1999).

[12] I. Terasaki, Y. Sasago and K. Uchinokura, Phys. Rev. B56 R12685 (1997).

[13] T. Motohashi, R. Ueda, E. Naujalis, T. Tojo, I. Terasaki, T. Atake, M. Karppinen and H.Yamauchi, Phys. Rev. B67 064406 (2003).

[14] K. Takada, H. Sakurai, E. Takayama-Muromachi, F. Izumi, R. A. Dilanian and T. Sasaki, Nature422 53 (2003).

[15] W. Koshibae, K. Tsutsui and S. Maekawa, Phys. Rev. B62 6869 (2000).

[16] K. Kusakabe and H. Aoki, Physica. B 194-196 217 (1994).

[17] W. Gill and D. J. Scalapino, Phys. Rev. B35 215 (1987) .

[18] J. Kuei and R. T. Scalettar, Phys. Rev. B55 14968 (1997) .

[19] J. E. Hirsch, Phys. Rev. B56 11022 (1997).

[20] K. Held and D. Vollhardt, Eur. Phys. J. B 5 473 (1998).

[21] S. Q. Shen, Phys. Rev. B57 6474 (1998).

[22] T. Momoi and K. Kubo, Phys. Rev. B58 R567 (1998).

[23] H. Sakamoto, T. Momoi and K. Kubo, Phys. Rev. B65 224403 (2002).

[24] K. Sano and Y. Ono, J. M. M. M. 310 (2) e319 (2007).

[25] H. C. Lee, P. Azaria and E. Boulat, Phys. Rev. B69 155109 (2004).

21

Ferromagnetism and Superconductivity in Two-band Hubbard Models

－21－

[26] T. Shirakawa, Y. Ohata, S. Nishimoto, J. M. M. M. 310 (2) 663 (2007)

[27] H. Tasaki, Prog. Theor. Phys. 99 489 (1998).

[28] J. Solyom, Adv. Phys. 28, 201 (1979).

[29] V. J. Emery, in Highly Conducting One-Dimensional Solids, edited by J. T. Devreese, R. Evrandand V. van Doren, (Plenum, New York, 1979), p.327.

[30] F.D.M. Haldane, J. Phys. C14 2585 (1981).

[31] J. Voit, Rep. Prog. Phys. 58 977 (1995).

[32] H. J. Schulz, Phys. Rev. Lett. 64 2831 (1990).

[33] M. Ogata, M. U. Luchini, S. Sorella and F. F. Assaad, Phys. Rev. Lett. 66 2388 (1991).

[34] K. Sano and Y. Ono, J. Phys. Soc. Jpn. 63 1250 (1994);J. Phys. Chem. Solids. 62 281 (2001);63 1567 (2002).

[35] L. Balentz and M.P.A. Fisher, Phys. Rev. B53 12133 (1996).

[36] M. Fabrizio, Phys. Rev, B 54 10054 (1996).

[37] V. J. Emery, S. A. Kivelson and O. Zachar, Phys. Rev. B59 15641 (1999).

[38] In this case, the electronic state is described by two-component charge degrees of freedom, whilethe single band model has a single component charge degree of freedom. The bosonization methodfor the two-chain model with a small interchain hopping[32, 35, 36, 37] shows that the model hasa spin-gaped phase and SC and CDW correlations decay as ∼ r

− 12Kρ and ∼ r−2Kρ respectively

(SDW correlations decay exponentially). Hence, SC correlation is dominant for Kρ > 0.5. In thiscase, one of the charge degree of freedom has a charge gap and the remain has gapless excitations.

[39] K. Sano, Physica B281&282 829 (2000); J. Phys. Soc. Jpn. 69 1000 (2000).

[40] K. Kusakabe, S. Watanabe and Y. Kuramoto, J. Phys. Soc. Jpn. Suppl. 71 311 (2002).

[41] K. Sano and Y. Ono, J. Phys. Soc. Jpn. 72 1847 (2003).

[42] K. Sano and Y. Ono, J. Phys.: Condens. Matter 19 14528 (2007).

[43] Detail of the Moebius boundary condition is discussed by T.F.A. Muller and T. M. Rice, Phys.Rev. B58 3425 (1998).

[44] R. Assaraf, P. Azaria, M. Caffarel and P. Lecheminant, Phys. Rev. B60, 2299 (1999).

[45] M. Imada, J. Phys. Soc. Jpn. 70 1218 (2001).

[46] K. Yamaji, J. Phys. Soc. Jpn. 70 1476 (2001).

[47] K. Sano and Y. Ono, Physica C205 (1993) 170;Phys. Rev. B51 (1995) 1175; Physica C242 113(1995).

[48] X. Zotos, W. Lehr and W. Weber, Z. Phys. B74 289 (1989).

[49] A. Sudbø, C.M. Varma, T. Giamarchi, E.B. Stechel and T. Scalettar, Phys. Rev. Lett. 70 978(1993).

[50] W. Barford and E. R. Gagliano Physica B194-196, 1455 (1994); see also, C. Vermeulen, W.Barford and E. R. Gagliano, Europhys. Lett. 28 653 (1994).

[51] E.B. Stechel, A. Sudbø, T. Giamarchi and C.M. Varma, Phys. Rev. B51 553 (1995).

[52] A. W. Sandvik and A. Sudbø, Phys. Rev. B54 R3746 (1996).

[53] K. Sano and Y. Ono, J. Phys. Soc. Jpn. 67 389 (1998);67 (1998) 4151.

22

Kazuhiro Sano and Yoshiaki Ōno

－22－

[54] W. Metzner and D. Vollhardt, Phys. Rev. Lett. 62 324 (1989).

[55] A. Georges, G. Kotliar, W. Krauth and M. J. Rozenberg, Rev. Mod. Phys. 68 13 (1996).

[56] Th. Pruschke, M. Jarrell and J.K. Freericks, Adv. Phys. 44 187 (1995).

[57] M. Jarrell, Phys. Rev. Lett. 69 168 (1992).

[58] M. Caffarel and W. Krauth, Phys. Rev. Lett. 72 1545 (1994).

[59] O. Sakai and Y. Kuramoto, Solid State Commun. 89 307 (1994).

[60] R. Bulla, Phys. Rev. Lett. 83 136 (1999).

[61] J.K. Freericks and M. Jarrell, Phys. Rev. Lett. 74, 186 (1995).

[62] M. Ulmke, Eur. Phys. J. B 1 301 (1998).

[63] T. Obermeier, T. Pruschke and J. Keller, Phys. Rev. B56 R8479 (1997).

[64] M. Jarrell and T. Pruschke, Z. Phys. B 90, 187 (1993).

[65] A. Koga, Y. Imai and N. Kawakami, Phys. Rev. B66 165107 (2002).

[66] Y. Ono, M. Potthoff and R. Bulla, Phys. Rev. B67 035119 (2003).

[67] A. Georges, G. Kotliar and W. Krauth, Z. Phys. B 92 313 (1993).

[68] T. Mutou, H. Takahashi and D. S. Hirashima, J. Phys. Soc. Jpn. 66 2781 (1997).

[69] Y. Ono and K. Sano, J. Phys. Chem. Solids 62 285 (2001).

[70] Y. Ono, R. Bulla and A. C. Hewson, Eur. Phys. J. B 19 375 (2001).

[71] Y. Ono, R. Bulla, A. C. Hewson and M. Potthoff, Eur. Phys. J. B 22 283 (2001).

[72] Y. Ono and K. Sano, J. Phys. Soc. Jpn. Suppl. 71 356 (2002).

[73] Y. Ohashi and Y. Ono, J. Phys. Soc. Jpn. 70 2989 (2001).

[74] Y. Ohashi and Y. Ono, J. Phys. Soc. Jpn. Suppl. 71 217 (2002).

[75] H. Watanabe and S. Doniach, Phys. Rev. B57 3829 (1998).

[76] In our previous paper[69], we reported that TSS is close to TTS and is slightly higher than TTS

for all n, but the calculation was done with a smaller system size Ns = 4.

[77] V. Berezinskii, JETP Lett. 20 287 (1974).

[78] A. Tahvildar-Zadeh, M. Hettler and M. Jarrell, Phil. Mag. B 78 365 (1998).

[79] H. Masuda, T. Fujita, T. Miyashita, M. Soda, Y. Yasui, Y. Kobayashi and M. Sato, J. Phys. Soc.Jpn. 72 873 (2003).

23

Ferromagnetism and Superconductivity in Two-band Hubbard Models

－23－